The CHSH or Bell inequalities are based on quite dubious assumptions that seem a poor match for how real hidden variables should behave and the current quantum mechanics is still full of inconsistencies.
Much of the classic quantum mechanics, e.g. based on the Schrödinger or Dirac equations, is compatible with local hidden variables and I have not seen yet a precise account of which are the more recent axioms/assumptions that introduce an incompatibility with local hidden variables.
Thus I do not believe that the non-existence of local hidden variables has been proven beyond any reasonable doubt.
Much of the classic quantum mechanics, e.g. based on the Schrödinger or Dirac equations, is compatible with local hidden variables and I have not seen yet a precise account of which are the more recent axioms/assumptions that introduce an incompatibility with local hidden variables.
Thus I do not believe that the non-existence of local hidden variables has been proven beyond any reasonable doubt.